The value of `1/5div3/10" of "4/9-4/5xx1(1)/(9)div8/15+3/4+3/4div1/2` is :

Options

A. `25/12`
B. `11/12`
C. `17/12`
D. `7/12`

Answer

To solve the expression \( \frac{1}{5} \div \frac{3}{10} \text{ of } \frac{4}{9} - \frac{4}{5} \times \frac{1}{9} \div \frac{8}{15} + \frac{3}{4} + \frac{3}{4} \div \frac{1}{2} \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert "of" to multiplication**:
The term "of" in mathematics indicates multiplication. Thus, we rewrite the expression:
\[ \frac{1}{5} \div \frac{3}{10} \times \frac{4}{9} - \frac{4}{5} \times \frac{1}{9} \div \frac{8}{15} + \frac{3}{4} + \frac{3}{4} \div \frac{1}{2} \]

2. **Calculate \( \frac{1}{5} \div \frac{3}{10} \)**:
To divide fractions, we multiply by the reciprocal:
\[ \frac{1}{5} \div \frac{3}{10} = \frac{1}{5} \times \frac{10}{3} = \frac{10}{15} = \frac{2}{3} \]

3. **Now calculate \( \frac{2}{3} \times \frac{4}{9} \)**:
\[ \frac{2}{3} \times \frac{4}{9} = \frac{8}{27} \]

4. **Calculate \( \frac{4}{5} \times \frac{1}{9} \div \frac{8}{15} \)**:
First, calculate \( \frac{4}{5} \times \frac{1}{9} \):
\[ \frac{4}{5} \times \frac{1}{9} = \frac{4}{45} \]
Now divide by \( \frac{8}{15} \):
\[ \frac{4}{45} \div \frac{8}{15} = \frac{4}{45} \times \frac{15}{8} = \frac{60}{360} = \frac{1}{6} \]

5. **Calculate \( \frac{3}{4} \div \frac{1}{2} \)**:
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2} \]

6. **Combine all parts**:
Now substitute back into the expression:
\[ \frac{8}{27} - \frac{1}{6} + \frac{3}{4} + \frac{3}{2} \]

7. **Find a common denominator**:
The least common multiple (LCM) of 27, 6, 4, and 2 is 108. Convert each fraction:
- \( \frac{8}{27} = \frac{32}{108} \)
- \( \frac{1}{6} = \frac{18}{108} \)
- \( \frac{3}{4} = \frac{81}{108} \)
- \( \frac{3}{2} = \frac{162}{108} \)

8. **Combine the fractions**:
\[ \frac{32}{108} - \frac{18}{108} + \frac{81}{108} + \frac{162}{108} = \frac{32 - 18 + 81 + 162}{108} = \frac{257}{108} \]

9. **Final answer**:
The value of the expression is:
\[ \frac{257}{108} \]